"I have a fever, but don't have a cough."
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In colloquial Mandarin 有 is sometimes used as a perfect marker similar to 了. But I associate that usage specifically with Taiwanese Mandarin (it's a Minnan borrowing, so maybe maybe it's also used like that in the area around Xiamen. Can't confirm that first-hand though). In Standard 普通话 you're correct.
Even if we take your premise that Taiwan isn't part of China, and therefore, somehow, anything that is said there is irrelevant, can you really speak for every region in Mainland China to conclude that this is not a correct sentence pattern?
There is a huge diversity of ways of speaking, just like there is in different parts of America, different parts of England, etc.
In the same way that learning to speak English isn't restricted to learning only the way that people speak in New York (or whatever), so this course is teaching a variety of correct grammar and vocabulary.
Here's a Chinese website which uses the phrase: https://k.sina.cn/article_5883288706_15eabdc8202000w9k4.html?from=health
That should help clear up the question of whether that sentence structure is used in China.